Physicists have long sought to describe the universe in terms of equations. Now, James Gates explainshow research on a class of geometric symbols known as adinkras could lead to fresh insights into thetheory of supersymmetry – and perhaps even the very nature of reality!

Transcript of "Symbols of power adinkras and the nature of reality by s james gates"

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physicsworld.com Volume 23 No 6 June 2010 Symbols of powerAdinkras and the nature of reality Cell control Fighting cancer with physicsFits and starts Why randomness does not rule our lives Who, what, when? Deciding upon a discovery

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Feature: Physics and geometry physicsworld.comSymbols of powerPhysicists have long sought to describe the universe in terms of equations. Now, James Gates explainshow research on a class of geometric symbols known as adinkras could lead to fresh insights into thetheory of supersymmetry – and perhaps even the very nature of realityJames Gates is a In the land of theoretical physics, equations have al- more complicated figures known as “adinkras”, atheoretical physicist ways been king. Indeed, it would probably be fair to name Faux suggested. The word “adinkra” is of Westat the University of caricature theoretical physicists as members of a com- African etymology, and it originally referred to visualMaryland, US, pany called “Equations-R-Us”, since we tend to view symbols created by the Akan people of Ghana and thee-mail gatess@ new equations as markers of progress. The modern era Gyamen of Côte d’Ivoire to represent concepts orwam.umd.edu of equation prediction began with Maxwell in 1861, aphorisms. However, the mathematical adinkras we continued through the development of Einstein’s equa- study are really only linked to those African symbols tions of general relativity in 1916 and reached its first by name. Even so, it must be acknowledged that, like peak in the 1920s with the Schrödinger and Dirac equa- their forebears, mathematical adinkras also represent tions. Then a second, postwar surge saw the develop- concepts that are difficult to express in words. Most ment of equations describing the strong force and the intriguingly, they may even contain hints of something electroweak force, culminating in the creation of the more profound – including the idea that our universe Standard Model of particle physics in about 1973. The could be a computer simulation, as in the Matrix films. equations trend continues today, with the ongoing struggle to create comprehensive equations to describe If you knew SUSY like we know SUSY… superstring theory. This effort – which aims to incor- To understand what adinkras are, we must first exam- porate the force of gravity into physical models in a way ine the physical theory to which they relate: supersym- that the Standard Model does not – marks the extant metry, commonly abbreviated as SUSY. The concept boundary of a long tradition. of symmetry is ubiquitous in nature, but on a more Yet equations are not the only story. To an extent, technical level it has been a powerful mathematical geometrical representations of physical theories have tool for the development of equations. Einstein recog- also been useful when correctly applied. The most fa- nized that there was a symmetry between the effects mous incorrect geometrical representation in physics observed by someone in an accelerating spacecraft far is probably Johannes Kepler’s model of planetary or- away from all planets and those observed by someone bits; initially, Kepler believed the orbits could be des- standing on the planet’s surface. He called this recog- cribed by five regular polygons successively embedded nition the “happiest thought” of his life, and he used it within each other, but he abandoned this proposition to determine the form of his equations of general re- when more accurate data became available. A less well lativity, which describe how matter warps space and known but much more successful example of geometry time to create gravity. applied to physics is Murray Gell-Mann’s “eightfold Moving on to the Standard Model, the set of equa- way”, which is a means of organizing subatomic par- tions used to describe the physics of quarks, leptons ticles. This organization has an underlying explanation (the family of particles that contains the electron) and using triangles with quarks located at the vertices. force-carrying particles like the photon (carrier of the For the past five years, I and a group of my col- electromagnetic force) is also largely determined by leagues (including Charles Doran, Michael Faux, Tris- symmetry groups. Photons, for example, possess a type tan Hubsch, Kevin Iga, Greg Landweber and others) of symmetry known as U(1), which means that two dis- have been following the geometric-physics path pion- tinct photons can produce the same electric and mag- eered by Kepler and Gell-Mann. The geometric ob- netic forces on a charged particle. Another important jects that interest us are not triangles or octagons, but symmetry is the SU(3) symmetry of quarks, which can be visualized using what mathematicians call a “weight- space diagram” (figure 1). This diagram shows theLike their African forebears, entire family of nuclear particles of which the proton, p, and neutron, n, are members. The location of particlesmathematical adinkras represent in this diagram is determined by particle properties called isospin and strangeness, the values of which wereconcepts that are difficult to express first measured in the 1950s and 1960s. Six triangles lurk inside it – you can see them if you draw lines from thein words, and they may even contain centre to each vertex – and this “triangular” symmetry is part of what leads to the designation SU(3). Such diagrams are more than pictures. In fact, ithints of something more profound was an insight drawn from such diagrams that led34 Physics World June 2010

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physicsworld.com Feature: Physics and geometry Mat WardComplex ideas, complex shapes Adinkras – geometric objects that encode mathematical relationships between supersymmetric particles – are named after symbols thatrepresent wise sayings in West African culture. This adinkra is called “nea onnim no sua a, ohu”, which translates as “he who does not know can become knowledgeablethrough learning”.Gell-Mann and George Zweig to a new understanding symmetries. When we know that certain symmetries areof nuclear matter. Gell-Mann and Zweig realized that present in nature, we can use one experiment to predictpatterns in diagrams showing families of nuclear par- the outcome of many others.ticles meant that those particles must be made up As its name implies, the theory of supersymmetryof smaller, more fundamental particles: quarks. The takes the idea of symmetry a step further. In the Stan-nuclear-particle octet diagram gets its name because dard Model there is a dichotomy between leptons andthere are particles associated with each of its six vertices, quarks – collectively called “matter particles” – andand two additional particles associated with its centre, force-carrying particles like photons. All matter parti-hence an “octet” of particles. This diagram is useful as a cles are fermions, particles with half-integer quantumkind of accounting tool: in certain nuclear reactions, spin that obey the Pauli exclusion principle. Force-two or more experiments will lead to simply related carrying particles, in contrast, are bosons, which haveresults if one member of this family is replaced by integer spin and can violate the exclusion principle.another. For example, measuring how a proton is de- This means that not only photons but also gluonsflected from a neutron by the strong nuclear force will (which carry the strong nuclear force), the W and Zyield a result that is directly related to the deflection of bosons (which carry the weak nuclear force), and evena Σ– particle from a neutron. This is the power of using the hypothetical Higgs boson are all free to possess anyPhysics World June 2010 35

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Feature: Physics and geometry physicsworld.com1 Weight-space diagrams 2 From squares to adinkras n p Φ2 D2 D1 Ψ1 Ψ2 Σ0 Σ– Σ+ D1 D2 Λ Φ1 Ψ1 Ψ2 D1Φ1 = i Ψ1 D2Φ1 = i Ψ2 D2 D1Φ2 = i Ψ2 D2Φ2 = – i Ψ1 Ξ– Ξ0 D1 D1 D Ψ = ∂ Φ D Ψ = – ∂ Φ 1 1 T 1 2 1 T 2This weight-space diagram shows the “baryon octet” group of D2 D1Ψ2 = ∂T Φ2 D2Ψ2 = ∂T Φ1particles, including the proton (p), neutron (n) and six more exoticspecies known as hyperons. Particles are arranged according to their A square can be transformed into two distinct adinkras.isospin (how they interact with the strong nuclear force) and the The set of eight super-differential equations relates to thenumber of strange quarks they contain (their “strangeness”). Φ1 Φ2 bottom adinkra. allowed quantum numbers in composite systems. by additional operations. Each of these decorations has SUSY breaks this rule that all matter particles are a mathematical significance, which I will discuss later. fermions and all carriers are bosons. It does this by For the moment, let us just concentrate on building a relating each Standard Model particle to a new form simple adinkra. of matter and energy called a “superpartner”. In its To make a square into an adinkra, we begin by pla- simplest form, SUSY states that every boson has a cing a white dot at one vertex (figure 2). The rules of corresponding “super-fermion” associated with it, and adinkras then dictate that the two line segments con- vice versa. These superpartners have not yet been ob- nected to the white dot must have black dots at their served in nature, but one of the main tasks of CERN’s opposite ends. This means that the final unpopulated Large Hadron Collider (LHC) will be to look for ex- vertex is connected to “black dot” vertices, so it must perimental evidence of their existence. If the LHC be populated by a white dot. Next, we need to assign finds these superpartners, then the Standard Model directions to each line segment, or link. To keep track of will have to be replaced by the Minimal Supersym- these different directions, we assign distinct colours to metric Standard Model (MSSM), or perhaps another each of them: all links that point in the same direction more exotic variant. are assigned the same colour, and links that point in From the point of view of equations, however, SUSY different directions are never assigned the same colour. presents an additional challenge. Even if the LHC finds Then, we need to assign an “edge-parity” to each evidence that we live in a supersymmetric universe, link: each coloured line can be drawn as either solid or there are many different sets of equations that incor- dashed. Every two-colour closed path in an adinkra porate supersymmetry. How, then, do we pick the right must contain an odd number of dashed links. One last ones? The answer, of course, is that we pick the equa- rule is that white dots and black dots are never allowed tions that agree with experimental observations. How- to have the same vertical position; that is, no black dot ever, we can also ask a more subtle question: how do in an adinkra is ever allowed to appear at the same we ensure that the SUSY property is made manifest at height as a white dot. Figure 2 shows a square that has every stage of calculations involving the quantum be- been “decorated” in two different ways and made into haviour of these equations? It is here that adinkras two distinct adinkras. might prove useful. Just as a weight-space diagram is a There is no limit to the number of colours that may graphical representation that precisely encodes the be used to construct an adinkra. As a result, higher- mathematical relations between the members of SU(3) dimensional adinkras have a certain aesthetic appeal families, so an adinkra is a graphical representation (figure 3). As Einstein once said, “After a certain high that precisely encodes the mathematical relations be- level of technical skill is achieved, science and art tend tween the members of supersymmetry families. to coalesce in aesthetics, plasticity and form.” Perhaps the “artistic” depictions shown here are an example Building up adinkras of this. Now that we know a little bit about how adinkras can be But adinkras, like Gell-Mann’s octets, are not just used, we can begin to discuss what they look like. All pictures. In fact, they are in some ways rather similar adinkras are constructed by starting with squares, cubes to Feynman diagrams, which are the series of line draw- and their higher-dimensional generalizations; these ings used to describe calculations in quantum electro- structures provide a “skeleton” that is then “decorated” dynamics. Like Feynman diagrams, adinkras are a36 Physics World June 2010

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physicsworld.com Feature: Physics and geometryprecise mathematical description of calculations. They 3 Multidimensional adinkrasalso serve as an aid to performing these calculations, Gregory D Landwebersince the way that adinkras are constructed provides astreamlined description of the most compact sets ofequations with the SUSY property. But while Feynmandiagrams describe calculations for particle quantumbehaviour, adinkras are connected instead to mathe-matical objects known as Clifford algebras and super-differential equations. Clifford algebras were introduced by the Englishmathematician and philosopher William KingdomClifford in the 1870s as mathematical constructionsthat generalize complex numbers. However, they alsoprovide the mathematical basis for our modern under-standing of fermions. Where adinkras are concerned, ifone ignores the information contained in the verticalheight of the same type of dots in an adinkra, then thatadinkra provides an exact description of mathematicalmatrices associated with Clifford algebras. For ex-ample, using the rules associated with adinkras, the bot-tom adinkra in figure 2 yields two of the three “Paulimatrices” (elements of a Clifford algebra) that describethe spin states of fermions. A second connection to mathematics is even moresimilar to Feynman diagrams. It can be shown that eachadinkra corresponds to a distinct set of super-differ-ential equations. Super-differential equations involveboth the ordinary derivative operator (invented by These large n-colour folded adinkras represent complex systems of super-differential equations.Newton and Leibnitz) and a newer type of operatorcalled a “super derivative”, which was invented in themid-1970s by the mathematician Felix Berezin and within each diagram: whenever the “starting” dot isthen elaborated on by the physicists Abdus Salam and higher in the adinkra than the “target” dot, this ordin-John Strathdee. Super derivatives, represented by the ary derivative appears on the right-hand side of the cor-links in an adinkra, are similar to the ordinary deriv- responding equation. The dashed links simply insertative, except that they are allowed to violate the usual minus signs into some equations. You should haveproduct rule for derivatives. The super-differential enough information now to apply this analysis to theequations for the bottom adinkra derived from a second diagram in order to write down its associatedsquare are shown in figure 2. equations – although, in time-honoured fashion, I have Since there are only two types of coloured links, there left this as an exercise for the reader.are only two super derivatives: D1 associated with greenlinks and D2 associated with red links. We also have two SUSY and adinkrasbosonic superfunctions (Φ1 and Φ2) associated with the Returning now to the concept of supersymmetry,correspondingly labelled white dots and two fermionic Salam and Strathdee devised a simple test to deter-superfunctions (Ψ1 and Ψ2) associated with the corres- mine when systems of equations possess the propertypondingly labelled black dots. As complex numbers of SUSY. The system shown in figure 2 easily passesgenerally consist of both a real and imaginary part, a Salam and Strathdee’s test, but this does not neces-superfunction consists of both bosonic and fermionic sarily mean that they are the equations that theoristsparts. To turn these components of the adinkra into a from the Equations-R-Us company are seeking. Inset of equations, we begin by picking one dot – let’s use fact, they are not: aside from the Pauli matrices, thethe bottom-left one as an example – and writing its square-derived adinkras are just too simple to be asso-associated superfunction, Φ1, to the left of an equal ciated with differential equations that have physicalsign. Next, we choose one of the coloured links and meaning. The same is true for adinkras based on a 3Dwrite its associated D to the left of the superfunction. cube. However, with a 4D hypercube, or tesseract, itFor the green link this would be D1; for the red link it is a different story. The four-colour adinkra (figure 4)would be D2. Then we look to see what dot is at the demonstrates a behaviour that is not present forother end of this link. If we pick the green link, the adinkras with fewer colours: it can be broken into two“target dot” is the one associated with the superfunc- separate, smaller adinkras. These smaller adinkrastion Ψ1, so this symbol belongs on the right of the equals do have physical meaning. The one on the far right issign. These rules alone are enough to give us the upper in fact related to Maxwell’s equations. If one firstfour equations in figure 2. removes the uppermost open dot and then performs To “derive” the second group of four equations we the Salam–Strathdee test, then Maxwell’s equationsneed to introduce the ordinary differential operation, involving current charges emerge. Similarly, remov-denoted by Ѩ⌻. The manner in which it appears in the ing the two uppermost dots from the centre adinkraequations is controlled by the relative height of the dots followed by the Salam–Strathdee test leads to thePhysics World June 2010 37

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Feature: Physics and geometry physicsworld.com 4 Adinkras within adinkras + The “decorated tesseract” adinkra on the left can be broken down into two separate adinkras. The author’s collaboration of mathematicians and other physicists has introduced the name “gnomoning” for this process of subtracting a smaller adinkra from larger ones. The word gnomoning was used by Euclid, the founder of geometry, to describe a plane figure obtained by removing a smaller figure that is similar to the larger one. equations for the behaviour of the electron and its wanted changes to a transmitted signal. Hamming’s SUSY partner (known as the “selectron”). idea was for the sending computer to insert extra bits Some of the equations described here have been into words in a specific manner such that the receiving known for some time to physicists who study SUSY. computer could, by looking at the extra bits, detect and Yet it was not until 2009 that research on adinkras correct errors introduced by the transmission process. (arXiv:0902.3830) showed that these geometric ob- His algorithm for the insertion of these extra bits is jects can mimic the behaviour of the equations, and known as the “Hamming code”. The construction of thus provided the first evidence that adinkras could be such error-correcting codes has been pursued since the related to physics. The next key question to answer is beginning of the computer age and many different whether the reverse process can also occur – beginning codes now exist. These are typically divided into famil- with an adinkra and using it to derive, via a set of well- ies; for example, the “check-sum extended Hamming defined rules, something like the Maxwell or Dirac code” is a rather complicated variant of the Hamming equations. In 2001 (arXiv:hep-th/0109109) my stu- code and it belongs to a family known as “doubly even dents and I conjectured that this could indeed be the self-dual linear binary error-correcting block codes” case, but only if we could encode the properties of (an amazing mouthful!). Yet whatever family they be- 4D equations onto objects in a mathematical 1D for- long to, all error-correction codes serve the same func- mat. Though this conjecture has not yet been proven, tion: they are used to detect errors and allow the correct work completed by Faux, Iga and Landweber in 2009 transmission of digital data. (arXiv:0907.4543, arXiv:0907.3605) has provided the How does this relate to adinkras? The middle adink- strongest evidence to date of its correctness. So, just ra in figure 4 is obtained by folding the image on the as weight-space diagrams opened a new way to con- left of the figure. The folding involves taking pairs of ceptualize the physics of nuclear matter, it is conceiv- the dots of the same type and “fusing them together” able that adinkras may yield an entirely new way to as if they were made of clay. In general, an adinkra- formulate theories that possess the property of SUSY. folding process will lead to diagrams where the associ- ated equations do not possess the SUSY property. In From theoretical physics to codes order to ensure that this property is retained, we must As it turns out, it is not just four-colour adinkras that carry out the fusing in such a way that white dots are can be separated into two smaller adinkras with the only fused with other white dots, black dots with other same number of colours; adinkras with more than four black dots, and lines of a given colour and dashing are colours also possess this property of separability. But only joined with lines that possess the same properties. why does this occur only for four or more colours? In- Most foldings violate this, but there is one exception – vestigating this question launched our “treasure hunt” and it happens to be related to a folding that involves in a completely unexpected direction: computer codes. doubly even self-dual linear binary error-correcting Modern computer and communication technologies block codes. have come to prominence by transmitting data rapidly The adinkra in figure 5 is the same as the left-hand and accurately. These data consist principally of strings part of figure 4 but for simplicity it is shown without of ones and zeros (called bits) written in long sequences dashed edges. We pick the bottom dot as a startingAdinkras may called “words”. When these computer words are trans- point and assign it an address of (0000). To move to anyyield an mitted from a source to a receiver, there is always the of the dots at the second level requires traversing one ofentirely new chance that static noise in the system can alter the con- the coloured links. There are four distinct ways in whichway to tent of any word. Hence, the transmitted word might this can be done. To move to any dot at the third levelformulate arrive at the receiver as pure gibberish. from the bottom dot requires the use of two different One of the first people to confront this problem was coloured links, and so on for the rest of the adinkra. Intheories that the mathematician Richard Hamming, who worked on this way, every dot is assigned an address, from (0000)possess the the Manhattan Project during the Second World War. to (1111). These sequences of ones and zeros are binaryproperty of In 1950 he introduced the idea of “error-correcting computer words.supersymmetry codes” that could remove or work around any un- To accomplish the folding that maintains the SUSY38 Physics World June 2010

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physicsworld.com Feature: Physics and geometryproperty in the associated equations, we must begin 5 Coded adinkrasby squeezing the bottom dot together with the upperdot. When their addresses are added bit-wise to oneanother, this yields the sequence (1111). If we con- (1111)tinue this folding process, always choosing pairs ofdots so that their associated “words” sum bit-wise to(1111), we can transform the adinkra on the left-handside of figure 4 to the one on the right. Thus, main- (1110) (1101) (1011) (0111)taining the equations’ SUSY property requires thatthe particular sequence of bits given by (1111) be usedin the folding process. The process used to meet thiscriterion happens to correspond to the simplest mem-ber of the family containing the check-sum extended (1100) (1001) (0110) (1010) (0101) (0011)Hamming code. The part of science that deals with the transmissionof data is called information theory. For the most part,this is a science that has largely developed in ways thatare unrelated to the fields used in theoretical physics.However, with the observation that structures from (1000) (0100) (0010) (0001)information theory – codes – control the structure ofequations with the SUSY property, we may be cross-ing a barrier. I know of no other example of this par-ticular intermingling occurring at such a deep level. (0000)Could it be that codes, in some deep and fundamentalway, control the structure of our reality? In asking this The decorated-tesseract adinkra and its associated computer “words”. For simplicity, thequestion, we may be ending our “treasure hunt” in a adinkra is shown without dashed lines.place that was anticipated previously by at least onepioneering physicist: John Archibald Wheeler. craziness from it one after another, like lifting layersLife in the Matrix? off an onion, at the heart of the idea you will oftenWheeler, who died in 2008, was an extremely well- find a powerful kernel of truth.” Indeed, another ofregarded figure within physics. He served as advisor to Wheeler’s “crazy” ideas – his suggestion that a positrona clutch of important physicists, including Richard can be treated as an electron moving backwards in timeFeynman, while his own work included the concept – played a role in Feynman later winning a Nobel prize.of the “S-matrix” (a mathematical tool that helps us As for my own collaboration on adinkras, the pathunderstand Standard Model particles). Beyond the my colleagues and I have trod since the early 2000s hasphysics community, Wheeler is probably best known led me to conclude that codes play a previously unsus-for coining the terms “black hole” and “wormhole”. pected role in equations that possess the property ofBut he also coined a slightly less familiar phrase – “it supersymmetry. This unsuspected connection suggestsfrom bit” – and this is what concerns us here. that these codes may be ubiquitous in nature, and could The idea of “it from bit” is a complex one, and even be embedded in the essence of reality. If this is theWheeler’s own description of it is probably still the best. case, we might have something in common with theIn 1990 he suggested that “every ‘it’ – every particle, Matrix science-fiction films, which depict a world whereevery field of force, even the space–time continuum everything human beings experience is the product ofitself – derives its function, its meaning, its very exist- a virtual-reality-generating computer network.ence entirely…from the apparatus-elicited answers to If that sounds crazy to you – well, you could be right.yes-or-no questions, binary choices, bits”. The “it from It is certainly possible to overstate mathematical linksbit” principle, he continued, “symbolizes the idea that between different systems: as the physicist Eugeneevery item of the physical world has at bottom…an Wigner pointed out in 1960, just because a piece ofimmaterial source and explanation: that which we call mathematics is ubiquitous and appears in the descrip-reality arises in the last analysis from the posing of tion of several distinct systems does not necessarilyyes–no questions and the registering of equipment- mean that those systems are related to each other. Theevoked responses; in short, that all things physical number π , after all, occurs in the measurement of cir-are information-theoretic in origin and that this is a cles as well as in the measurement of population dis-participatory universe”. tributions. This does not mean that populations are When I first heard the idea of “it from bit” as a young related to circles.physicist, I thought Wheeler must be crazy. The con- Yet for a moment, let us imagine that this alternativecept of a world made up of information just sounded Matrix-style world contains some theoretical physicists,strange, and (although I did not know it at the time) I and that one of them asks, “How could we discoverwas not the only one who thought so. However, some- whether we live inside a Matrix?”. One answer mighttimes crazy ideas turn out to be true, and Wheeler has be “Try to detect the presence of codes in the laws thatbeen proved right before. As Feynman said, “When I describe physics.” I leave it to you to decide whetherwas [Wheeler’s] student, I discovered that if you take Wigner’s warning should be applied to the theoreticalone of his crazy ideas and you unwrap the layers of physicists living in the Matrix – and to us. ■Physics World June 2010 39